An Analytical Approach to Evaluating Bivariate Functions of
Fuzzy Numbers with One Local Extremum
Arthur Seibel and Josef Schlattmann
Workgroup on System Technologies and Engineering Design Methodology
Hamburg University of Technology, 21073 Hamburg, Germany
Keywords:
Parameter Uncertainties, Bivariate Functions, Fuzzy Numbers, Analytical Fuzzy Calculus.
Abstract:
This paper presents a novel analytical approach to evaluating continuous, bivariate functions of independent
fuzzy numbers with one local extremum. The approach is based on a parametric α-cut representation of fuzzy
numbers and allows for the inclusion of parameter uncertainties into mathematical models.
1 INTRODUCTION
There is an increasing effort in the scientific commu-
nity to provide suitable methods for the inclusion of
uncertainties into mathematical models. One way to
do so is to introduce parametric uncertainty by rep-
resenting the uncertain model parameters as fuzzy
numbers (Dubois and Prade, 1980) and evaluating the
model equations by means of Zadeh’s extension prin-
ciple (Zadeh, 1975). The evaluation of this classical
formulation of the extension principle, however, turns
out to be a highly complex task (Klimke, 2006). For-
tunately, Buckley and Qu (1990) provide an alterna-
tive formulation that operates on α-cuts and is appli-
cable to continuous functions of independent fuzzy
numbers. Powerful numerical techniques have been
developed to implement this alternative formulation
(Moens and Hanss, 2011). However, there is no gen-
eral analytical approach for a calculus with fuzzy
numbers. For this purpose, a practical analytical ap-
proach to evaluating continuous, monotonic functions
of independent fuzzy numbers was introduced by the
authors (Seibel and Schlattmann, 2013, 2014), which
is based on the alternative formulation of the exten-
sion principle. In this paper, we extend this approach
to bivariate functions of fuzzy numbers with one local
extremum and no saddle points.
An outline of this paper is as follows. In Section
2, we give a definition of fuzzy numbers and present
two important types. In Section 3, we briefly recall
Zadeh’s extension principle and introduce the alter-
native formulation based on α-cuts. In Section 4, we
describe our analytical approach and give two illustra-
tive examples. Finally, in Section 5, some conclusions
are drawn.
2 FUZZY NUMBERS
Fuzzy numbers (Dubois and Prade, 1980) are a spe-
cial class of fuzzy sets (Zadeh, 1965), which can be
defined as follows.
A normal, convex fuzzy set ˜x over the real line
R is called fuzzy number if there is exactly one ¯x ∈R
with µ
˜x
( ¯x) = 1 and the membership function is at least
piecewise continuous. The value ¯x is called the modal
or peak value of ˜x.
Theoretically, an infinite number of possible types
of fuzzy numbers can be defined. However, only few
of them are important for engineering applications
(Hanss, 2005). These typical fuzzy numbers shall be
described in the following.
2.1 Triangular Fuzzy Numbers
Due to its very simple, linear membership function,
the triangular fuzzy number (TFN) is the most fre-
quently used fuzzy number in engineering. In order
to define a TFN with the membership function
µ
˜x
(x) =
1 +
x − ¯x
τ
L
, ¯x −τ
L
≤ x ≤ ¯x,
1 −
x − ¯x
τ
R
, ¯x < x ≤ ¯x + τ
R
,
(1)
we use the parametric notation (Hanss, 2005)
˜x = tfn( ¯x, τ
L
,τ
R
),
where ¯x denotes the modal value, τ
L
denotes the left-
hand, and τ
R
denotes the right-hand spread of ˜x (cf.
Figure 1). If τ
L
= τ
R
, the TFN is called symmetric. Its
α-cuts x(α) =
x
L
(α),x
R
(α)
result from the inverse
89
Seibel A. and Schlattmann J..
An Analytical Approach to Evaluating Bivariate Functions of Fuzzy Numbers with One Local Extremum.
DOI: 10.5220/0005026500890094
In Proceedings of the International Conference on Fuzzy Computation Theory and Applications (FCTA-2014), pages 89-94
ISBN: 978-989-758-053-6
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
¯x −τ
L
¯x
¯x + τ
R
0
1
x
µ
˜x
(x)
Figure 1: Triangular fuzzy number.
functions of Eqs. (1) with respect to x:
x
L
(α) = ¯x −τ
L
(1 −α), 0 < α ≤1,
x
R
(α) = ¯x + τ
R
(1 −α), 0 < α ≤1.
2.2 Gaussian Fuzzy Numbers
Another widely-used fuzzy number in engineering is
the Gaussian fuzzy number (GFN), which is based on
the normal distribution from probability theory. In or-
der to define a GFN with the membership function
µ
˜x
(x) =
exp
−
1
2
x − ¯x
σ
L
2
, x ≤ ¯x,
exp
−
1
2
x − ¯x
σ
R
2
, x > ¯x,
we use the parametric notation (Hanss, 2005)
˜x = gfn( ¯x, σ
L
,σ
R
),
where ¯x denotes the modal value, σ
L
denotes the left-
hand, and σ
R
denotes the right-hand standard devia-
tion of ˜x (cf. Figure 2). If σ
L
= σ
R
, the GFN is called
symmetric. Its α-cuts x(α) =
x
L
(α),x
R
(α)
result in
x
L
(α) = ¯x −σ
L
p
−2ln(α), 0 < α ≤ 1,
x
R
(α) = ¯x + σ
R
p
−2ln(α), 0 < α ≤ 1.
3 EXTENSION PRINCIPLE
Zadeh’s extension principle (Zadeh, 1975) allows for
extending any real-valued function to a function of
fuzzy numbers. More specifically, let ˜x
1
,. . . , ˜x
n
be n
independent or noninteractive fuzzy numbers, and let
f : R
n
→ R be a function with y = f (x
1
,. . . , x
n
). The
fuzzy extension ˜y = f ( ˜x
1
,. . . , ˜x
n
) is then defined by
µ
˜y
(y) = sup
y= f (x
1
,...,x
n
)
min{µ
˜x
1
(x
1
),. . ., µ
˜x
n
(x
n
)}.
In case of interdependency between ˜x
1
,. . . , ˜x
n
, the
minimum operator should be replaced by a suitable
¯x
0
1
√
e
1
σ
L
σ
R
x
µ
˜x
(x)
Figure 2: Gaussian fuzzy number.
triangular norm (Scheerlinck, 2011). In this paper,
however, we restrict ourselves to independent fuzzy
numbers.
The evaluation of this classical formulation of the
extension principle turns out to be a highly complex
task (Klimke, 2006). Fortunately, Buckley and Qu
(1990) provide an alternative formulation that oper-
ates on α-cuts:
Let x
1
(α),. . ., x
n
(α) denote the α-cuts of the n in-
dependent fuzzy numbers ˜x
1
,. . . , ˜x
n
, and let f be con-
tinuous. Then, the α-cuts y(α) =
y
L
(α),y
R
(α)
of ˜y
can be computed from
y
L
(α) = min{f (x
1
,. . . , x
n
)|(x
1
,. . . , x
n
) ∈ Ω(α)},
y
R
(α) = max{f (x
1
,. . . , x
n
)|(x
1
,. . . , x
n
) ∈ Ω(α)},
where Ω(α) = x
1
(α) ×···× x
n
(α) represent the n-
dimensional interval boxes that are spanned by the α-
cuts x
1
(α),. . ., x
n
(α).
If the continuous function f is (strictly) monotonic
increasing in x
i
, i = 1, .. . ,k, and (strictly) monotonic
decreasing in x
j
, j = 1,. . ., `, in the domain of inter-
est, and if k + ` = n, then, the minimum values of f
inside of every sub-domain Ω(α) are always found at
the left boundaries of x
i
(α) and the right boundaries
of x
j
(α), and its maximum values at the right bound-
aries of x
i
(α) and the left boundaries of x
j
(α), respec-
tively. In such case, the α-cuts y(α) =
y
L
(α),y
R
(α)
of ˜y become (Seibel and Schlattmann, 2013)
y
L
(α) = f
x
L
i
(α),x
R
j
(α)
, 0 < α ≤1,
y
R
(α) = f
x
R
i
(α),x
L
j
(α)
, 0 < α ≤1,
(2)
with x
m
(α) =
x
L
m
(α),x
R
m
(α)
, m = 1,. . ., n. If Eqs.
(2) are invertible with respect to α, then the mem-
bership function of ˜y yields (Seibel and Schlattmann,
2013)
µ
˜y
(y) =
(
y
L
(α)
−1
, y
L
(0) < y ≤y
L
(1),
y
R
(α)
−1
, y
R
(1) < y < y
R
(0).
The analytical approach, which is presented in the
next section, is based on this alternative formulation
of the extension principle.
FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
90
4 ANALYTICAL APPROACH
For evaluating continuous, nonmonotonic functions
of independent fuzzy numbers, Dong and Shah (1987)
and Fortin et al. (2008) suggest to include the extreme
points as constant profiles into the computation. How-
ever, this is not enough and can lead to erroneous re-
sults, as was pointed out by Wood et al. (1992). More
specifically, all permutations of the interval bound-
aries of x
m
(α), m = 1,.. . ,n, with the components of
the extreme points have to be considered as well.
Let f be a continuous, bivariate function with
positive arguments and no saddle points. If x
∗
=
(x
∗
1
,x
∗
2
) is the only maximum (minimum) of f , then
f (x −x
∗
) is (strictly) monotonic decreasing (increas-
ing) in x
1
and x
2
. In order to compute the α-cuts
y(α) =
y
L
(α),y
R
(α)
of ˜y, we distinguish between
the following two scenarios.
4.1 x
∗
is a Maximum
The modal point
¯
x = ( ¯x
1
, ¯x
2
) divides the (positive) do-
main of interest into four subdomains. The analytical
solution depends in which subdomain the maximum
is located. Let α
∗
1
= µ
˜x
1
(x
∗
1
) and α
∗
2
= µ
˜x
2
(x
∗
2
), then
1. x
∗
1
≥ ¯x
1
and x
∗
2
≥ ¯x
2
:
y
L
(α) = f
x
L
1
(α),x
L
2
(α)
, 0 < α ≤ 1,
(a) α
∗
1
≥ α
∗
2
:
y
R
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
2
,
f
x
∗
1
,x
R
2
(α)
, α
∗
2
< α ≤ α
∗
1
,
f
x
R
1
(α),x
R
2
(α)
, α
∗
1
< α ≤ 1,
(b) α
∗
1
≤ α
∗
2
:
y
R
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
1
,
f
x
R
1
(α),x
∗
2
, α
∗
1
< α ≤ α
∗
2
,
f
x
R
1
(α),x
R
2
(α)
, α
∗
2
< α ≤ 1,
2. x
∗
1
≤ ¯x
1
and x
∗
2
≥ ¯x
2
:
y
L
(α) = f
x
R
1
(α),x
L
2
(α)
, 0 < α ≤ 1,
(a) α
∗
1
≥ α
∗
2
:
y
R
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
2
,
f
x
∗
1
,x
R
2
(α)
, α
∗
2
< α ≤ α
∗
1
,
f
x
L
1
(α),x
R
2
(α)
, α
∗
1
< α ≤ 1,
(b) α
∗
1
≤ α
∗
2
:
y
R
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
1
,
f
x
L
1
(α),x
∗
2
, α
∗
1
< α ≤ α
∗
2
,
f
x
L
1
(α),x
R
2
(α)
, α
∗
2
< α ≤ 1,
3. x
∗
1
≤ ¯x
1
and x
∗
2
≤ ¯x
2
:
y
L
(α) = f
x
R
1
(α),x
R
2
(α)
, 0 < α ≤ 1,
(a) α
∗
1
≥ α
∗
2
:
y
R
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
2
,
f
x
∗
1
,x
L
2
(α)
, α
∗
2
< α ≤ α
∗
1
,
f
x
L
1
(α),x
L
2
(α)
, α
∗
1
< α ≤ 1,
(b) α
∗
1
≤ α
∗
2
:
y
R
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
1
,
f
x
L
1
(α),x
∗
2
, α
∗
1
< α ≤ α
∗
2
,
f
x
L
1
(α),x
L
2
(α)
, α
∗
2
< α ≤ 1,
4. x
∗
1
≥ ¯x
1
and x
∗
2
≤ ¯x
2
:
y
L
(α) = f
x
L
1
(α),x
R
2
(α)
, 0 < α ≤ 1,
(a) α
∗
1
≥ α
∗
2
:
y
R
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
2
,
f
x
∗
1
,x
L
2
(α)
, α
∗
2
< α ≤ α
∗
1
,
f
x
R
1
(α),x
L
2
(α)
, α
∗
1
< α ≤ 1,
(b) α
∗
1
≤ α
∗
2
:
y
R
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
2
,
f
x
R
1
(α),x
∗
2
, α
∗
1
< α ≤ α
∗
2
,
f
x
R
1
(α),x
L
2
(α)
, α
∗
2
< α ≤ 1.
Note that f (x
∗
1
,x
∗
2
) is only necessary if (x
∗
1
,x
∗
2
) lies
within the domain of interest.
The solution paths of y(α) for ˜x
1
and ˜x
2
being of
triangular type are illustrated in Figure 3. We can see
that, starting from the extremum, which is marked by
the yellow circle, the modal point acts as an attractor
and pulls the solution parallel to a coordinate axis di-
rectly into one of the two principal diagonals running
from the lower-left to the upper-right corner and from
the lower-right to the upper-left corner of the domain
supp( ˜x
1
) ×supp( ˜x
2
), respectively.
Example 1. The function f
1
: R
2
+
→ R with
y
1
= f
1
(x
1
,x
2
) = −x
2
1
−x
2
2
+ 5x
1
+ x
2
shall be evaluated for the two fuzzy numbers ˜x
1
=
tfn(2,2, 3) and ˜x
2
= tfn(2,2, 2).
Since f
1
has only one maximum at x
∗
= (2.5, 0.5),
x
∗
is located in subdomain 4, and α
∗
1
= 0.8
¯
3 ≥0.25 =
α
∗
2
, the α-cuts y
1
(α) =
y
L
1
(α),y
R
1
(α)
of ˜y
1
are
y
L
1
(α) = −8α
2
+ 24α −12, 0 < α ≤1,
y
R
1
(α) =
6.5, 0 < α ≤0.25,
−4α
2
+ 2α + 6.25, 0.25 < α ≤0.8
¯
3,
−13α
2
+ 17α, 0.8
¯
3 < α ≤1.
AnAnalyticalApproachtoEvaluatingBivariateFunctionsofFuzzyNumberswithOneLocalExtremum
91
x
1
x
2
0
0.25
0.5
0.75
1
α
∗
1
≥ α
∗
2
α
∗
1
≤ α
∗
2
µ
(a) Subdomain 1.
x
1
x
2
0
0.25
0.5
0.75
1
α
∗
1
≥ α
∗
2
α
∗
1
≤ α
∗
2
µ
(b) Subdomain 2.
x
1
x
2
0
0.25
0.5
0.75
1
α
∗
1
≥ α
∗
2
α
∗
1
≤ α
∗
2
µ
(c) Subdomain 3.
x
1
x
2
0
0.25
0.5
0.75
1
α
∗
1
≥ α
∗
2
α
∗
1
≤ α
∗
2
µ
(d) Subdomain 4.
Figure 3: Solution paths of y(α).
With y
L
1
(0) = −12, y
L
1
(1) = 4 = y
R
1
(1), y
R
1
(0.8
¯
3) ≈
5.1, and y
R
1
(0.25) = 6.5, the membership function of
˜y
1
yields
µ
˜y
1
(y) =
3
2
−
1
4
p
12 −2y, −12 < y ≤4,
17
26
+
1
26
p
289 −52y, 4 < y ≤5.1,
1
4
+
1
4
p
26 −4y, 5.1 < y < 6.5.
The plot of µ
˜y
1
(y) is shown in Figure 4.
4.2 x
∗
is a Minimum
The modal point
¯
x = ( ¯x
1
, ¯x
2
) divides the (positive) do-
main of interest into four subdomains. The analytical
solution depends in which subdomain the minimum
is located. Let α
∗
1
= µ
˜x
1
(x
∗
1
) and α
∗
2
= µ
˜x
2
(x
∗
2
), then
1. x
∗
1
≥ ¯x
1
and x
∗
2
≥ ¯x
2
:
y
R
(α) = f
x
L
1
(α),x
L
2
(α)
, 0 < α ≤ 1,
(a) α
∗
1
≥ α
∗
2
:
y
L
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
2
,
f
x
∗
1
,x
R
2
(α)
, α
∗
2
< α ≤ α
∗
1
,
f
x
R
1
(α),x
R
2
(α)
, α
∗
1
< α ≤ 1,
(b) α
∗
1
≤ α
∗
2
:
y
L
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
1
,
f
x
R
1
(α),x
∗
2
, α
∗
1
< α ≤ α
∗
2
,
f
x
R
1
(α),x
R
2
(α)
, α
∗
2
< α ≤ 1,
2. x
∗
1
≤ ¯x
1
and x
∗
2
≥ ¯x
2
:
y
R
(α) = f
x
R
1
(α),x
L
2
(α)
, 0 < α ≤ 1,
(a) α
∗
1
≥ α
∗
2
:
y
L
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
2
,
f
x
∗
1
,x
R
2
(α)
, α
∗
2
< α ≤ α
∗
1
,
f
x
L
1
(α),x
R
2
(α)
, α
∗
1
< α ≤ 1,
(b) α
∗
1
≤ α
∗
2
:
y
L
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
1
,
f
x
L
1
(α),x
∗
2
, α
∗
1
< α ≤ α
∗
2
,
f
x
L
1
(α),x
R
2
(α)
, α
∗
2
< α ≤ 1,
3. x
∗
1
≤ ¯x
1
and x
∗
2
≤ ¯x
2
:
y
R
(α) = f
x
R
1
(α),x
R
2
(α)
, 0 < α ≤ 1,
(a) α
∗
1
≥ α
∗
2
:
y
L
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
2
,
f
x
∗
1
,x
L
2
(α)
, α
∗
2
< α ≤ α
∗
1
,
f
x
L
1
(α),x
L
2
(α)
, α
∗
1
< α ≤ 1,
(b) α
∗
1
≤ α
∗
2
:
y
L
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
1
,
f
x
L
1
(α),x
∗
2
, α
∗
1
< α ≤ α
∗
2
,
f
x
L
1
(α),x
L
2
(α)
, α
∗
2
< α ≤ 1,
FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
92
−12 −8.3
−4.6
−0.9 2.8
6.5
0
0.25
0.5
0.75
1
y
µ
˜y
(y)
Figure 4: Membership function of ˜y
1
.
4. x
∗
1
≥ ¯x
1
and x
∗
2
≤ ¯x
2
:
y
R
(α) = f
x
L
1
(α),x
R
2
(α)
, 0 < α ≤ 1,
(a) α
∗
1
≥ α
∗
2
:
y
L
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
2
,
f
x
∗
1
,x
L
2
(α)
, α
∗
2
< α ≤ α
∗
1
,
f
x
R
1
(α),x
L
2
(α)
, α
∗
1
< α ≤ 1,
(b) α
∗
1
≤ α
∗
2
:
y
L
(α) =
f
x
∗
1
,x
∗
2
, 0 < α ≤α
∗
1
,
f
x
R
1
(α),x
∗
2
, α
∗
1
< α ≤ α
∗
2
,
f
x
R
1
(α),x
L
2
(α)
, α
∗
2
< α ≤ 1.
Note that, again, f (x
∗
1
,x
∗
2
) is only necessary if
(x
∗
1
,x
∗
2
) lies within the domain of interest.
In this scenario, the solution paths of y(α) are the
same as those in Figure 3.
Example 2. Now, the function f
2
: R
2
+
→ R with
y
2
= f
2
(x
1
,x
2
) = −f
1
(x
1
,x
2
) = x
2
1
+ x
2
2
−5x
1
−x
2
shall be evaluated for the two fuzzy numbers from Ex-
ample 1.
Since f
2
has only one minimum at x
∗
= (2.5, 0.5),
x
∗
is located in subdomain 4, and α
∗
1
= 0.8
¯
3 ≥0.25 =
α
∗
2
, the α-cuts y
2
(α) =
y
L
2
(α),y
R
2
(α)
of ˜y
2
are
y
L
2
(α) =
−6.5, 0 < α ≤0.25,
4α
2
−2α −6.25, 0.25 < α ≤0.8
¯
3,
13α
2
−17α, 0.8
¯
3 < α ≤1,
y
R
2
(α) = 8α
2
−24α + 12, 0 < α ≤1.
With y
L
2
(0.25) = −6.5, y
L
2
(0.8
¯
3) ≈ −5.1, y
L
2
(1) =
−4 = y
R
2
(1), and y
R
2
(0) = 12, the membership func-
tion of ˜y
2
yields
µ
˜y
2
(y) =
1
4
+
1
4
p
4y + 26, −6.5 < y ≤−5.1,
17
26
+
1
26
p
52y + 289, −5.1 < y ≤−4,
3
2
−
1
4
p
2y + 12, −4 < y < 12.
−6.5
−2.8 0.9
4.6
8.3 12
0
0.25
0.5
0.75
1
y
µ
˜y
(y)
Figure 5: Membership function of ˜y
2
.
The plot of µ
˜y
2
(y) is shown in Figure 5. Note that
µ
˜y
7
(y) and µ
˜y
8
(y) are symmetric to each other.
5 CONCLUSIONS
We extended our analytical approach from Seibel and
Schlattmann (2013, 2014) to bivariate functions of
fuzzy numbers with one local extremum and no sad-
dle points. It is based on an alternative formulation
of the extension principle and allows for the inclusion
of parameter uncertainties into mathematical models.
Using the patterns from Figure 3, this approach can be
easily extended to bivariate functions with more than
one extremum, see Wood et al. (1992) and Degrauwe
(2007) for similar numerical approaches.
An analytical solution has the advantage that the
degrees of membership of the fuzzy output can be
computed for any value within the support, whereas
a numerical solution only provides a finite number of
values. Furthermore, our approach also allows a sym-
bolic processing of uncertainties.
In further research activities, this approach shall
be extended to general, nonmonotonic functions of
independent fuzzy numbers, where the influence of
interdependency shall be investigated as well.
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